Dimension after multiple convolutions in ConvNets
The 64 here is the number of filters that are used.
The picture is kind of misleading in that it leaves out the transition of the maxpool.


Below is a text description of the size of the features as they go through the network with the number of filters in bold.


The first 2 layers in the diagram you posted contain 64 3x3 convs
resulting in a 224x224x64 matrix of features. 
This is then fed into a maxpool which reduces the size to a 112x112x64 matrix. 
This is then fed to 3 layers of 128 3x3 convs resulting in a 112x112x128
matrix. 
Then another maxpool giving a 56x56x128 matrix.
Feeding that to 3 layers of 256 3x3 convs results in a 56x56x256 matrix.
This is then fed into another maxpool giving a 28x28x256 matrix,
Which is then fed into 3 layers of 512 3x3 convs resulting in a
28x28x512 matrix.
Another maxpool gives a 14x14x512 matrix which is fed to 3 layers of 512 3x3 convs giving a matrix of 14x14x512 features.
Another maxpool reduces this to a 7x7x512 which is then
given to 3 fully connected layers of 4096 units each before being
sent to a softmax.